Celebratio Mathematica

David H. Blackwell

A Tribute to David Blackwell

by Murray Rosenblatt

I had only oc­ca­sion­al con­tact with Dav­id Black­well
through the years. But I al­ways found him to be
a warm, gra­cious per­son with a friendly greet­ing.
He entered the Uni­versity of Illinois at Urb­ana–Cham­paign
in 1935 at the age of six­teen and
re­ceived a bach­el­or’s de­gree in math­em­at­ics in
1938 and a mas­ter’s de­gree in 1939. Black­well
wrote a doc­tor­al thes­is on Markov chains with
Joseph L. Doob
as ad­visor in 1941. Two earli­er
al­most-con­tem­por­ary doc­tor­al stu­dents of J. L.
Doob were
Paul Hal­mos,
with a doc­tor­al de­gree
in 1938, and
War­ren Am­brose,
with the de­gree in
1939.

Black­well was a postdoc­tor­al fel­low at the In­sti­tute
for Ad­vanced Study for a year from 1941
(hav­ing been awar­ded a Ros­en­wald fel­low­ship).
There was an at­temp­ted ra­cist in­ter­ven­tion by the
then-pres­id­ent of Prin­ceton, who ob­jec­ted to the
hon­or­if­ic des­ig­na­tion of Black­well as a vis­it­ing fel­low
at Prin­ceton (all mem­bers of the In­sti­tute had
this des­ig­na­tion). He was on the fac­ulty of Howard
Uni­versity in the math­em­at­ics de­part­ment from
1944 to 1954.
Ney­man
sup­por­ted the ap­point­ment
of Dav­id Black­well at the Uni­versity of Cali­for­nia,
Berke­ley, in 1942, but this fell through due to
the pre­ju­dices at that time (see
[e4]).
However, in
1955 Dav­id Black­well was ap­poin­ted pro­fess­or of
stat­ist­ics at UC Berke­ley and be­came chair of the
de­part­ment the fol­low­ing year.

Black­well wrote over ninety pa­pers and made
ma­jor con­tri­bu­tions in many areas — dy­nam­ic pro­gram­ming,
game the­ory, meas­ure the­ory, prob­ab­il­ity
the­ory, in­form­a­tion the­ory, and math­em­at­ic­al
stat­ist­ics. He was an en­ga­ging per­son with broad-ran­ging
in­terests and deep in­sights. He was
quite in­de­pend­ent but of­ten car­ried out re­search
with oth­ers. In­ter­ac­tion with
Gir­shick
prob­ably
led him to re­search on stat­ist­ic­al prob­lems of
note. Re­searches with
K. Ar­row,
R. Bell­man,
and
E. Barankin
fo­cused on game the­ory. Joint work
with
A. Thomasi­an
(a stu­dent of his) and
L. Breiman
was on cod­ing prob­lems in in­form­a­tion the­ory. He
also car­ried out re­searches with col­leagues at UC
Berke­ley, such as
Dav­id Freed­man,
Lester Du­bins,
J. L. Hodges,
and
Peter Bick­el.
The
Rao–Black­well
the­or­em deal­ing with the ques­tion of op­tim­al
un­biased es­tim­a­tion is due to him.

He was elec­ted the first Afric­an Amer­ic­an mem­ber
of the Na­tion­al Academy of Sci­ences, USA,
and re­ceived many oth­er awards. He was a dis­tin­guished
lec­turer. We’re thank­ful that he sur­vived
the dif­fi­culties that Afric­an Amer­ic­ans had to en­dure
in a time of great bi­as (in his youth). He was a
per­son of sin­gu­lar tal­ent in the areas of stat­ist­ics
and math­em­at­ics.

The prob­ab­il­ity meas­ure \( P \) on \( \mathcal{A} \) is called
ex­treme if \( P(A) = 0 \) or 1 for all \( A \in \mathcal{A} \). An \( \mathcal{A} \)-atom
is the in­ter­sec­tion of all ele­ments of \( \mathcal{A} \) that con­tain
a giv­en point of \( \Omega \). If for \( A \in \mathcal{A},\, P(A) = 1 \), \( P \) is
said to be sup­por­ted by \( A \).

Then we have:

As­sume \( \mathcal{B} \) is count­ably gen­er­ated. Then
each of the con­di­tions im­plies the suc­cessor.

There is an ex­treme count­ably ad­dit­ive
prob­ab­il­ity meas­ure on \( \mathcal{A} \) that is sup­por­ted
by no \( \mathcal{A} \)-atom be­long­ing to \( \mathcal{A} \).

This res­ult shows that, for \( \Omega \) the in­fin­ite product
of a sep­ar­able met­ric space con­tain­ing more
than one point, neither the tail field, the field of
sym­met­ric events, nor the in­vari­ant field ad­mit
a prop­er r.c.d (reg­u­lar con­di­tion­al dis­tri­bu­tion).
They weak­en the count­able ad­dit­iv­ity con­di­tion of
an r.c.d. to fi­nite ad­dit­iv­ity and add (1) to ob­tain
the no­tion of a nor­mal con­di­tion­al dis­tri­bu­tion and
ar­rive at suf­fi­cient con­di­tions for ex­ist­ence. Later
re­lated re­search by
Berti
and
Rigo
[e3]
con­siders the
r.c.d.s with ap­pro­pri­ate weak­en­ings of the concept
of prop­er.

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